Millikan Oil Drop

Millikan’s Oil-Drop Experiment
(MW DVD 1081 )
The determination of the charge on the electron by Robert Millikan established
him as one of the great physicists of all time, and in 1923 he was awarded the Nobel
Prize for physics for this work. The determination of this fundamental unit of charge
showed conclusively that the charge on any body cannot be indefinitely reduced, and that
a charged body may carry a whole number of units but never a fraction of a unit.
J.J. Thomson, on measuring the ratio of e/m for cathode rays, assumed that the
charge on the electron was always the same. In later experiments of 1903, H.A. Wilson,
in the Cavendish Laboratory at Cambridge (England), attempted to measure the charge
on the electron by observing the rate of fall of the top surface of an ionized cloud of water
vapor placed between two horizontal plates. He then set up an electrical field between
the plates, with the positive voltage at the bottom plate. The top of the cloud fell faster.
From these measurements, Wilson’s value of e ranged between 0.7x10 -19 C and 1.5x10 -19
C.
Millikan, at the University of Chicago, attempted to repeat the Wilson
measurement. He noticed that the individual droplets of water vapor remained suspended
in the electric field when the top plate was at a positive potential. This, in 1909, was the
start of Millikan’s work on a “balanced drop” method of determine e, which gave a value
of 1.56x10 -19 C (R.A. Millikan, Philosophical Magazine, 19, Feb 1910, p.209). Millikan
then realized that the problems caused by evaporation of the water drops could be
overcome if minute oil or mercury drops were used. He then designed his oil-drop
apparatus to attempt to measure the value to an accuracy of 0.1%. In 1910, in the journal
Science (32, Sept. 30, 1910, p.437) and in Physical Review (32, 1911, p.349), Millikan
reported the results of these experiments.
In the diagram, A and B represent two horizontal plates carefully machined to be
flat to within 1/50 of a
millimeter, measuring
20 cm in diameter and
placed 1.6 cm apart.
The electric field
between the plates
could be varied
between 3000 and
8000 volts per cm.
A light source
with a heat filter
illuminated the droplets from the side through a glass window. The droplets were
produced by an atomizer, and some found their way through the hole H into the chamber.
Once an oil drop was selected for observation, the hole H was closed. The droplets were
observed with a low-power microscope through another glass window, X rays were used
to change the charge on the droplets by ionizing the air. Millikan discovered that the
value of e appeared to be greater for smaller drops than for larger ones and concluded that
a correction to Stokes’ law was necessary for the viscous force on droplets of a size
comparable to the mean free path of the air molecules. Later papers by Millikan
(Physical Review 2, 1913, p.109 and Philosophical Magazine 6, 1917, p.34) discussed
this correction in great detail and described more elaborate versions of the apparatus,
including temperature control. One of the most crucial external measurements was of the
viscosity of air.

In our experiments, a modified version of Millikan’s apparatus is used. An
atomizer produces a large number of oil droplets between the plates. By selective
application of an electric field, one droplet is singled out for observation. The droplets
are contained in a glass-walled compartment, with horizontal plates only about 3 mm
apart and applied voltages in the 100 to 350-volt range. Fields lower than Millikan’s are
used since the droplets are also smaller than his. The droplets are observed with a highpower
microscope and a high-sensitivity closed-circuit television system.
Theory:
(I) Measurement of the mass and radius of the droplet.
Consider a spherical drop of density ρ and radius a falling under gravity in air of
density σ. Stokes’ law states that for a spherical body falling with velocity v in a
medium of viscosity η, the viscous resistive force is given by 6πηav. The
Archimedean upward force on the body will be 4πa 3 σg/3. If the falling body has
reached its terminal velocity v and is therefore no longer accelerating, then the
weight of the body must equal the upward force plus the viscous force.
Therefore,
3 3
4πa ρg 4πa σg
= 6πηav
+
3 3
weight viscous upward force
of drop force due to air
4 3
and ( ) 6
3 π a ρ − σ g = πηav
.
Typically, σ is close to ρ/1000 and therefore, within the accuracy of this
4 3
experiment, can be ignored. Then 6
3 π a ρ g = πηav
and 9η
v
a = .
2ρ
g
(II) Static Experiment
If an electric field E is now applied so that the drop, if charged, remains
suspended between the two plates, then the weight of the drop will be balanced by
the upward electric force. Thus if e’ is the total charge on the drop, then
eV '
mg = e'
E = s
where V s is the voltage across the plates and d is the plate
d
mgd ⎛4
3 ⎞ d
separation, and e'
= , or e'
= ⎜ πa ρ ⎟ g . Substituting in our equation above
V
⎝ ⎠ V
to eliminate a we get
s
3
s
1/2
⎧⎪
⎛ 9η
⎞ ⎫⎪v
Kv
e' = ⎨6πµ
d⎜
⎟ ⎬ =
⎪⎩
⎝ ρ g ⎠ ⎪⎭
V V
3/2 3/2
2
s s
where K is the constant within the brackets.
(III) Dynamic Experiment
If the applied voltage is increased above V s , the droplet will rise in this applied
field. If V is now the applied voltage, and the droplet has reached its terminal
velocity, then
3 3
4 π a ρ g eV ' 4 a g
+ 6πηav2
= +
π σ
3 d 3
weight viscous electric upward force
of drop force field force due to air
Solving this for e’, as for the static case, gives the equation:
,

1/2
Kv (
1+
v2)
v1
e'
= , where v 1 is the terminal velocity when falling, and v 2 is the
V
terminal velocity when rising. K is the same constant as for the static case.
(IV) Stokes’ Law Corrections
As a result of the oil-drop experiment, Millikan investigated the deviation from
Stokes’ law for small droplets comparable in size to the mean free path of air
molecules. The terminal velocity under these conditions is higher than that given
by Stokes’ law and a Stokes-Cunningham correction is applied.
Since the mean free path of air molecules is related to the atmospheric pressure,
the corrected viscous force can be written as
Method:
⎛ b ⎞
viscous force = 6πηav⎜1 + ⎟ .
⎝ pa ⎠
In Millikan’s original paper, b was equal to 6.18x10 -4 where p, the
atmospheric pressure, is measured in cm of mercury and a in cm.
This correction produces a modification to the equations for e’, requiring
3/2
⎛ b ⎞
them to be multiplied by the factor ⎜1+
⎟ .
⎝ pa ⎠
For the experimental conditions and radius of droplets shown in the
program, this factor is 0.83 on average, and the values of e’ should all be
multiplied by 0.85. Exact values, if required, can be found using the Stokes-
Cunningham correction given above in each case.
(1) Free fall under gravity
The velocity v 1 of a drop falling under gravity is found by measuring the
time t g for it to fall through a distance of 3.x10 -4 m after reaching its terminal
velocity. This measurement is repeated several times for the same drop to
average out the effects of Brownian motion. The scale shown on the television
monitor has been previously calibrated with the use of a gradicule placed in front
of the microscope. Each division corresponds to 0.1 mm.
(2) Static Experiment
The voltage V s applied to the plates is adjusted so that the drop remains
stationary in the field if view, V s is then measured. Now, using the values listed
below for K and t g (to calculate v 1 ) and substituting for K, V s , and v 1 in the static
equation will give the value of e’.
Results
(3) Dynamic Experiment
The velocity v 2 is found by measuring the time t f for the oil drop to rise through a
distance of 3x10 -4 m under a measured voltage V. Using the listed values of t g
and t f (to calculate v 1 and v 2 ) together with K and V, the value of e’ can be found
from the dynamic equation.
−1
Parameters used in the experiment
Separation between plates (d)
= 3.3x10 -3 m

Timed distance = 3.0x10 -4 m
Viscosity of air = 1.82x10 -5 Ns/m 2 at 20°C
Density of air = 1.29 kg/m 3
Density of oil = 9.67x10 2 kg/m 3
Atmospheric Pressure of air = 76.1 cm Hg
Gravitational Acceleration (g) = 9.81 m/s 2
Calculate value of K
= 1.05x10 -10 SI Units
Ancillary Data
Plate separation measurement
Mass of S.G. Bottle
Mass of S.G. Bottle & Water
Mass of S.G. Bottle & Oil
Static experiment
Drop # t g (avg) V s
1-5 In film In film
6 10.05 s 82 V
7 3.48 s 150 V
8 2.35 s 170 V
9 5.24 s 111 V
10 8.99 s 53 V
1 st reading: 122.07 mm
2 nd reading: 118.77 mm
17.396 g
41.572 g
40.776 g
Dynamic Experiments
Drop # t g (avg) t f (avg) V
1-4 In film In film in film
5 2.78 s 32.00 s 324 V
6 10.05 s 9.10 s 178 V
7 3.48 s 7.23 s 222 V
8 2.35 s 6.91 s 233 V
9 5.24 s 9.75 s 180 V
10 5.27 s 15.10 s 330 V
Conclusions
The calculated values of e’, even without Stokes’ law correction, will show that, if
the average lowest values of e’ are divided into the significantly larger values, the larger
values will be approximately simple multiples of the lowest value. Therefore, the
average lowest value corresponds to the fundamental unit charge e. The accurate value
for e within the experimental errors can be found by applying the Stokes’ law correction
to the results.
Student Activities: Student should be prepared to note for themselves the readings for the
first 5 static experiments and the first 4 dynamic experiments. A stopwatch will be
necessary.